Riemann zeta functions for Krull monoids (Q6556215)

The authors extend the notion of the classical Riemann zeta function, which is viewed as a function associated to the monoid of positive integers, to the more general setting of Krull monoids with torsion class groups.\N\NMonoids are assumed to be commutative, cancellative and torsion free, which means that the groupification (``the smallest'' group containing the monoid) is a torsion free abelian group. A monoid is \emph{reduced} if the group of units is trivial. Every monoid can be transformed into a reduced monoid by dividing by the group of units. For the purpose in the article there is no loss of generality in assuming that monoids are also reduced.\N\NIn addition to the notion of atoms (indescomposible elements) and prime elements, there is also the notion of strong atoms. These are atoms whose powers enjoy a unique factorization as a product of atoms (the obvious one). In unique factorization monoids these three notions coincide. For general monoids these concepts might be different. This phenomenon is illustrated with examples in Section 3.\N\NA \emph{divisor theory} is a monoid homomorphism \(\tau:M\to F\), where \(F\) is a free commutative monoid, satisfying the following conditions: (1) if \(b,c\in M\) and \(\tau(b)|\tau(c)\) in \(F\) then \(b|c\) in \(M\), (2) for every \(d\in F\) there exist \(b_1,\ldots,b_n\in M\) with \(d=\operatorname{gcd}\{\tau(b_1),\ldots,\tau(b_n)\}\). A \emph{Krull monoid} is a monoid \(M\) possessing a divisor theory \(\tau:M\to F\). In this case quotient group \(F/\tau(M)\) is well-defined and is called the \emph{class group} of \(M\). For example, an integral domain is a Krull domain if and only if its (reduced) multiplicative monoid is a Krull monoid.\N\NIf \(M\) is a (reduced) Krull monoid with torsion class group then the Decay Theorem [\textit{U. Krause} and \textit{C. Zahlten}, Mitt. Math. Ges. Hamb. 12, No. 3, 681--696 (1991; Zbl 0756.20010)] states that the monoid generated by the set of strong atoms \(\mathscr{S}(M)\) is a unique factorization monoid and every element in \(M\) has a power in \(\langle \mathscr{S}(M)\rangle\). A \emph{scale} in \(M\) is a multiplicative monoid homomorphism \(\sigma:M\to\mathbb{R}^\times\) such that \(\sigma(a)>1\) for every \(a\in \mathscr{S}(M)\). By using the Decay theorem the authors show first that to give a scale is the same as to give a function \(f:\mathscr{S}(M)\to \mathbb{R}_{>1}\). Then they define the Riemann zeta function of \(M\) at a scale \(\sigma\) as the series\N\[\N\zeta_M(\sigma)=\sum_{x\in \langle \mathscr{S}(M)\rangle}\frac{1}{\sigma(x)}\N\]\NIn their main result they show: (1) \(\zeta_M(\sigma)\) converges if and only if the series \(\sum_{a\in \mathscr{S}(M)}\frac{1}{\sigma(x)}\) converges, and (2) when \(\zeta_M(\sigma)\) converges it has a factorization as an Euler product \(\zeta_M(\sigma)=\prod_{a\in \mathscr{S}(M)}\frac{1}{1-(\sigma(a))^{-1}}\).\N\NThey obtain as an application that \(\mathscr{S}(M)\) is infinite if and only if \(\zeta_M(\sigma)\) diverges for some scale \(\sigma\). Moreover, in that case \(\sum_{a\in \mathscr{S}(M)}\frac{1}{\sigma(x)}\) diverges. They also show that if \(\zeta_M(\sigma)\) converges for some scale, then \(M\) is a unique factorization monoid if and only if the series \(\sum_{x\in M}\frac{1}{\sigma(x)}\) admits an Euler factorization over the primes.\N\NThe classical Riemann zeta function becomes a particular case as follows. For each positive real \(s\) the function \(\sigma: n\mapsto n^s\) is a scale for the multiplicative monoid \(\mathbb{N}\) and the classical Riemann zeta function at \(s\) is the monoid Riemann zeta function at \(\sigma\).